# Uncertainty relations for underdamped Langevin dynamics

## TL;DR

This paper extends the thermodynamic uncertainty relation to underdamped Langevin systems, revealing that both entropy production and dynamical activity constrain current fluctuations, with numerical verification on specific models.

## Contribution

It introduces a new bound for underdamped Langevin dynamics that incorporates dynamical activity, unlike previous overdamped cases.

## Key findings

- The relative fluctuation of a current is constrained by entropy production and dynamical activity.
- Dynamical activity significantly influences the uncertainty bound in underdamped systems.
- Numerical verification confirms the derived inequalities for specific models.

## Abstract

A trade-off between the precision of an arbitrary current and the dissipation, known as the thermodynamic uncertainty relation, has been investigated for various Markovian systems. Here, we study the thermodynamic uncertainty relation for underdamped Langevin dynamics. By employing information inequalities, we prove that, for such systems, the relative fluctuation of a current at a steady state is constrained by both the entropy production and the average dynamical activity. We find that, unlike what is the case for overdamped dynamics, the dynamical activity plays an important role in the bound. We illustrate our results with two systems, a single-well potential system and a periodically driven Brownian particle model, and numerically verify the inequalities.